Zurich Dynamics Talks*Time: Wednesdays (biweekly) 13:00, Room: Y27H46 (Irchel Math)
The problem of understanding the invariant measures for the linear flow on compact translation surfaces (and correspondingly for interval exchange transformations) has been much studied since the 1970s and is by now very well understood. Very little is however known about infinite translation surfaces. I will talk about the simplest case of infinite translation surfaces, which are obtained as $\Z^m$-covers of compact translation surfaces, giving rise to skew-products over IETs. In the first part of the talk I will present the problem and the existing results, and state the main theorem, which gives a classification of ergodic measures for skew-products which are periodic under renormalisation. In the second part of the talk I will explain the idea of the proof, which is an application of a symbolic result of Aaronson, Nakada, Sarig and Solomyak.
Critical points play a crucial role in 1D dynamical systems as natural sources of non-linearity. They allow seemingly simple maps to have extremely rich and complicated behaviors. One of the simplest settings in which one can study the dynamical effects of critical points is the class of unimodal interval maps (a generalization of real quadratic polynomials). The dynamics of these maps is now very well understood, due to the celebrated renormalization theory of Sullivan, McMullen, Lyubich and Avila. The goal of this talk is to generalize this theory to a higher dimensional setting: namely, to 2D area-contracting diffeomorphisms. The talk will consist of two parts. In the first, I will give the motivations, examples, definitions and the statement of the main theorem (a priori bounds). In the second, I will briefly outline the proof of a priori bounds in 1D, then give a rough sketch of how the core argument can be adapted to the 2D case by combining classic 1D techniques (such as Denjoy lemma or Koebe distortion theorem) with elements from Pesin theory. This is based on joint work with S. Crovisier, M . Lyubich and E. Pujals.
Given a hyperbolic surface of infinite volume it is due to Patterson that there are no embedded eigenvalues of the Laplacian in the essential spectrum [1/4, \infty). In my talk I will explain a generalization of this result on locally symmetric spaces of higher rank. The main example of geometries where our theorem applies are quotients of higher rank symmetric spaces w.r.t. images of Anosov representations.
Since the work of Minkowski in the early twentieth century, the space of lattices has been a fundamental tool in the study of natural or rational numbers. Then, Margulis and his followers, in particular Dani, showed that methods from ergodic theory could be used very efficiently in that setting. More recently, Schmidt and Summerer started the "parametric geometry of numbers", which is a way to describe diagonal orbits in the space of lattices, using a simple combinatorial coding. The goal of this mini-course is to introduce the main concepts of parametric geometry of numbers, and to use them to study two problems going back to Jarnik and Schmidt: - (Jarnik) Given r>(n+1)/n, does there exist x in R^n such that the inequality |x-p/q|< q^(-r) has infinitely many solutions p/q in Q^n, but for all c<1, the inequality |x-p/q|< cq^(-r) has only finitely many solutions p/q in Q^n? (And what is the Hausdorff dimension of the set of such points?) - (Schmidt) Given an l-dimensional subspace x in R^d, for what values of r can one always find an l-dimensional rational subspace v in Q^d arbitrarily close to x and such that the distance to x satisfies d(v,x)< H(v)^(-r)? (Height and distance on Grassmann varieties will be defined in the first lecture.) |